Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations

TL;DR

This paper classifies finite-dimensional pointed Hopf algebras with abelian coradical, showing they are cocycle deformations of their associated graded Hopf algebras, and extends this to Nichols algebras of diagonal type.

Contribution

It provides a complete classification of such Hopf algebras and constructs explicit cocycle deformations for Nichols algebras of diagonal type.

Findings

Finite-dimensional pointed Hopf algebras are cocycle deformations of graded Hopf algebras.

Explicit families of Hopf algebras are constructed as cocycle deformations.

Classification up to isomorphism is achieved for these Hopf algebras.

Abstract

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type $V$ with a principal realization in the category of Yetter-Drinfeld modules of a cosemisimple Hopf algebra $H$ and such that the Nichols algebra $\mathfrak{B}(V)$ is finite-dimensional, thus presented by a finite set $\mathcal G$ of relations, we define a family of Hopf algebras $\mathfrak{u}(\boldsymbol\lambda)$, $\boldsymbol\lambda\in \Bbbk^{\mathcal G}$, which are cocycle deformations of $\mathfrak{B}(V)\# H$ and such that $\operatorname{gr}\mathfrak{u}(\boldsymbol\lambda)\simeq \mathfrak{B}(V)\# H$.